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Properties

Label 1-163-163.14-r0-0-0
Degree $1$
Conductor $163$
Sign $0.0727 + 0.997i$
Analytic cond. $0.756968$
Root an. cond. $0.756968$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.963 + 0.268i)2-s + (0.657 + 0.753i)3-s + (0.856 − 0.516i)4-s + (0.597 − 0.802i)5-s + (−0.835 − 0.549i)6-s + (−0.565 + 0.824i)7-s + (−0.686 + 0.727i)8-s + (−0.135 + 0.990i)9-s + (−0.360 + 0.932i)10-s + (−0.981 + 0.192i)11-s + (0.952 + 0.305i)12-s + (−0.286 + 0.957i)13-s + (0.323 − 0.946i)14-s + (0.996 − 0.0774i)15-s + (0.466 − 0.884i)16-s + (0.973 + 0.230i)17-s + ⋯
L(s)  = 1  + (−0.963 + 0.268i)2-s + (0.657 + 0.753i)3-s + (0.856 − 0.516i)4-s + (0.597 − 0.802i)5-s + (−0.835 − 0.549i)6-s + (−0.565 + 0.824i)7-s + (−0.686 + 0.727i)8-s + (−0.135 + 0.990i)9-s + (−0.360 + 0.932i)10-s + (−0.981 + 0.192i)11-s + (0.952 + 0.305i)12-s + (−0.286 + 0.957i)13-s + (0.323 − 0.946i)14-s + (0.996 − 0.0774i)15-s + (0.466 − 0.884i)16-s + (0.973 + 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0727 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0727 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.0727 + 0.997i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.0727 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6512078627 + 0.6054267108i\)
\(L(\frac12)\) \(\approx\) \(0.6512078627 + 0.6054267108i\)
\(L(1)\) \(\approx\) \(0.7879344812 + 0.3607908479i\)
\(L(1)\) \(\approx\) \(0.7879344812 + 0.3607908479i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (-0.963 + 0.268i)T \)
3 \( 1 + (0.657 + 0.753i)T \)
5 \( 1 + (0.597 - 0.802i)T \)
7 \( 1 + (-0.565 + 0.824i)T \)
11 \( 1 + (-0.981 + 0.192i)T \)
13 \( 1 + (-0.286 + 0.957i)T \)
17 \( 1 + (0.973 + 0.230i)T \)
19 \( 1 + (0.813 + 0.581i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.713 - 0.700i)T \)
31 \( 1 + (-0.993 - 0.116i)T \)
37 \( 1 + (0.893 + 0.448i)T \)
41 \( 1 + (0.856 + 0.516i)T \)
43 \( 1 + (-0.627 + 0.778i)T \)
47 \( 1 + (0.249 - 0.968i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.686 - 0.727i)T \)
67 \( 1 + (-0.875 + 0.483i)T \)
71 \( 1 + (0.987 + 0.154i)T \)
73 \( 1 + (-0.875 - 0.483i)T \)
79 \( 1 + (-0.790 + 0.612i)T \)
83 \( 1 + (-0.431 - 0.902i)T \)
89 \( 1 + (-0.981 - 0.192i)T \)
97 \( 1 + (0.925 + 0.378i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.15349162795178229845819777916, −26.45405884299282001222944331307, −25.68855977367631869808695980650, −25.0813799778534106977599003650, −23.8183331566067573067919389044, −22.67867002923123423098904078094, −21.29673017698665211745314231443, −20.35641062166449725288457706680, −19.571842497660431615535812457004, −18.509327423066392385595953582958, −18.02193993847147916794771894006, −16.91830755905704867266824955020, −15.63741349057010170893813886587, −14.43349102742540811856869434624, −13.30646740435935610844477191030, −12.47489129689864640322098176697, −10.86271056899082530156790611443, −10.11994024687881833182135348837, −9.08274690585709771666020371115, −7.5861712043652184891900047005, −7.22266573232303222460272168509, −5.882900183448163625321989608616, −3.18606907553544326432747397656, −2.70623102467497917423848084661, −0.97236631017563021377925683190, 1.83077772699294207496601300452, 3.00272880083803279288065112437, 5.0003673237834236861396946598, 5.90857452060650773094590607284, 7.63907333541824943134458950807, 8.63106925214555984335429762572, 9.609654519646047941242892789084, 9.968442126823150668125075399563, 11.57297153797323000443760911252, 12.89375438318379667908772204371, 14.233975729226408307025633248414, 15.30884725986915727316937126075, 16.234323489020539960753625260075, 16.77483849287389306853131521513, 18.21621777439821395224842973713, 19.10862813743457440916935852235, 20.0881318010385298417537263576, 21.11362830562105086966990869604, 21.57667493788054352690648043856, 23.32000595751774493153027287999, 24.61347755392109340761305218190, 25.318290305658821361587282725931, 25.970419648751046377896595605776, 26.90565483102055727799680602361, 27.99046297969411592352271069499

Graph of the $Z$-function along the critical line